United's board appear to have lost patience with Moyes in the wake of the latest 2-0 defeat at former club Everton, which left United 13 points adrift of the Premier League top four.
And Neville has urged United to come out and clarify the manager's position as soon as possible.
"I fear for him," Neville told Monday Night Football. "When these explosions happen and all the media outlets go wild usually it ends up inevitably in a manager getting sacked.
"I have to say Manchester United as a football club do not normally comment on anything but they are going to have to clarify this position quickly because this afternoon has been strange.
"I find it repulsive the way in which these rumours break. I don't like it and I'll never get used to it even though I suppose it's the world in which we live in.
"But I think it has to be clarified pretty quickly by the club and they might have to break ranks and do what they don't normally do and that's make a statement pretty quickly."
Neville was part of one of the most successful United sides in history, winning eight Premier League titles and two European Cups during his time with the club.
But he believes that United may be acting prematurely if they sack Moyes just one year into his six-year contract, insisting the Scot deserves time to prove himself.
"That's my honest belief," he said. "I've supported the club for 15-20 years and played for the club for 18 years, so I've been associated with the club for 30 years and that's what I have always believed.
"Alex Ferguson himself got time in the early years when it was difficult for him. So I genuinely believe that when you give a man a six-year contract he deserves the opportunity and the time both professionally and out of respect.
"I've always felt Manchester United should be different and hold themselves up as a club that stands against what is happening in the game.
"I don't like it when professional people come out with statements that a manager should be sacked," Neville added. "It's not something that you'll ever hear from me because it's a difficult job and something that I've never done.
"Like I say - for David Moyes - I do fear for him because of the explosion that has happened today, but I'm hoping the club will just come out and clarify it and maybe it won't be true."